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arXiv:cond-mat/9704049 · Journal of the Physical Society of Japan

Effects of Umklapp Scattering on Electronic States in One Dimension

Masakazu Murakami, Hidetoshi Fukuyama

Published Aug 15, 1997• 2 citations
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Abstract

The effects of Umklapp scattering on electronic states are studied in one spatial dimension at absolute zero. The model is basically the Hubbard model, where parameters characterizing the normal ($U$) and Umklapp ($V$) scattering are treated independently. The density of states is calculated in the t-matrix approximation by taking only the forward and Umklapp scattering into account. It is found that the Umklapp scattering causes the global splitting of the density of states. In the presence of sufficiently strong Umklapp scattering, a pole in the t-matrix appears in the upper half plane, signalling an instability towards the '$G/2-$pairing' ordered state ($G$ is the reciprocal lattice vector), whose consequences are studied in the mean field approximation. It turns out that this ordered state coexists with spin-density-wave state and also brings about Cooper-pairs. A phase diagram is determined in the plane of $V$ and electron filling $n$.

DOI
10.1143/jpsj.66.2399

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The effects of Umklapp scattering on electronic states are studied in one spatial dimension at absolute zero. The model is basically the Hubbard model, where parameters characterizing the normal (UUU) and Umklapp (VVV) scattering are treated independently. The density of states is calculated in the t-matrix approximation by taking only the forward and Umklapp scattering into account. It is found that the Umklapp scattering causes the global splitting of the density of states. In the presence of sufficiently strong Umklapp scattering, a pole in the t-matrix appears in the upper half plane, signalling an instability towards the 'G/2−G/2-G/2−pairing' ordered state (GGG is the reciprocal lattice vector), whose consequences are studied in the mean field approximation. It turns out that this ordered state coexists with spin-density-wave state and also brings about Cooper-pairs. A phase diagram is determined in the plane of VVV and electron filling nnn.
The one-dimensional Mott metal-insulator transition is a typical strong correlation effect triggered by the Umklapp scattering. However, in a physical system, the Umklapp scattering coexists with the normal scattering, including both forward and backward scattering, which conserves the total momentum of scattered electrons. Therefore, it is not easy to quantify the contribution of the Umklapp scattering in a Mott metal-insulator transition. To resolve this difficulty, we propose to explore these scattering processes separately. We study the contribution of each scattering process in the one-dimensional Hubbard model using the momentum-space density-matrix renormalization group (kDMRG) and bosonization methods. Our kDMRG calculation confirms that the Mott charge gap results from the Umklapp scattering, but the normal scattering processes strongly renormalize its value. Furthermore, we present a scaling analysis of the Mott charge gap in the bosonization theory and show that the interplay between the Umklapp and forward scattering dictates the charge dynamics in the half-filled Hubbard model.
The model Hamiltonian of the one-dimensional electron gas with different types of interaction (forward and backward scattering, Umklapp processes) is studied by assuming a sharp bandwidth cut-off. Exact equalities among different response functions are obtained for specific values of coupling constants from simple symmetry considerations. The ground-state energy derived in mean-field approximation and the asymptotic behaviour of response functions as calculated by various authors are discussed with respect to these exact relations. The agreement of phase diagrams as obtained with different methods is traced back to the underlying symmetry. Giving up this symmetry, e.g. by assuming different cut-offs above and below the Fermi energy, changes the phase diagram. Furthermore this allows one to construct a mean-field ground state where superconductivity and density waves coexist in a finite region of coupling space. If the electron-phonon interaction is also taken into account it follows that the charge-density wave instability (together with a periodic lattice distortion) is favoured.
We report results obtained for the phase diagram and critical behavior of the extended Hubbard model in one spatial dimension at half-filling. We show that umklapp scattering is responsible for the tricritical behavior. We calculate the critical exponents at the tricritical point, and along the critical line separating the charge- and spin-density-wave phases. In addition, using Monte Carlo simulations we estimate the position of the tricritical point to occur at approximately U=1.5 and V=3.1 in units of the hopping energy.
The role of Umklapp scattering for electrons on a lattice is studied. Using renormalisation group methods it is shown that the effective low energy interaction involves not only forward scattering, but also non-forward Umklapp scattering. We show that the latter can happen to almost all of the electrons even when they have a filling of around one-half. The bahavior of these Umklapp couplings is studied in the loop expansion. It is shown that they remain marginal to all loop orders. The discussion is given in the context of two-dimensional lattices, but generalisation to three-dimensions is mentioned. We also briefly discuss the possible consequences of this on Fermi liquid theory and on physical properties of such electronic systems.
The electrical resistivity due only to mutual Coulomb interaction of the d -dimensional lattice electron systems ( d ≥2) in the presence of Umklapp scattering processes is studied on the basis of the Fermi liquid theory. It is shown that normal scattering processes generally contribute to the resistivity if Umklapp scattering processes are present. This fact gives a natural explanation of the resistivity of the d =∞ Hubbard model which is determined only by the damping rate of quasi-particles, γ. In the case of d =2, however, it is shown that the resistivity is proportional to T 2 , T being the temperature, even when normal scattering processes, which give T 2 log T -contribution to γ, contribute to the resistivity. In the special case of the square lattice, where the transfer integrals are only between the nearest neighbors, the coefficient of the T 2 -term in the second order of the interaction is shown to diverge as (1/δ) log (1 / δ) when δ→0, δ being the doping rate away from half-filling.
An exact solution of the one-dimensional electron gas on a lattice is given for the special case in which two of the four coupling constants have particular values. It is shown that umklapp scattering has the same effect on the charge-density waves as backward scattering across the Fermi "surface" has on the spin-density waves and that the method of Luther and Emery can be used to solve this more general problem. For repulsive electron-electron interactions the umklapp scattering produces a gap in the charge-density wave spectrum and this appears for low-lying excitations when there is a half-filled band. The use of renormalization-group scaling to solve for general values of the coupling constants is discussed.